Formal factorization of higher order irregular linear differential operators

TL;DR

This paper presents a formal factorization method for higher order irregular linear differential operators, reducing the non-commutative problem to a commutative one using Newton diagrams, and draws parallels with system diagonalization.

Contribution

It introduces a reduction process from non-commutative to commutative factorization, linking differential operators to pseudopolynomial factorization via Newton diagrams.

Findings

Reduction of non-commutative to commutative factorization

Connection between operator factorization and pseudopolynomial factorization

Parallelism between operator factorization and system diagonalization

Abstract

We study the problem of decomposition (non-commutative factorization) of linear ordinary differential operators near an irregular singular point. The solution (given in terms of the Newton diagram and the respective characteristic numbers) is known for quite some time, though the proofs are rather involved. We suggest a process of reduction of the non-commutative problem to its commutative analog, the problem of factorization of pseudopolynomials, which is known since Newton invented his method of rotating ruler. It turns out that there is an "automatic translation" which allows to obtain the results for formal factorization in the Weyl algebra from well known results in local analytic geometry. Besides, we draw some (apparently unnoticed) parallelism between the formal factorization of linear operators and formal diagonalization of systems of linear first order differential equations.